--- jupytext: text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.11.5 kernelspec: display_name: Python 3 language: python name: python3 --- # Spatial Relations: Co-location ## Topics * Co-locational spatial relationship * Readings: Lloyd Chp. 5 ## Fundamental/Basic Spatial Relationships * Relationships between two attribute values * >, < * difference, ratio * Relationships between two locations (spatial relationships) * = (occupy the exact same space) * Co-locate (share some space, i.e., overlap in space) * Distance (no co-locate) * Mapping unit as location in practice ## Co-locational Relationships * Share some space * Qualitative co-locational relationship * Intersect, touch, contain, ..., disjoint * Independent of coordinates * Topological relationship * Quantitative co-locational relationship * Where is the shared (or not shared) space located? * Based on and derived from coordinates ## Qualitative Locational (Topological) Relationship * Locations that share some space * intersect, touch, contain, ... * Most GIS support some common spatial relationships * Hard to define precisely * intersect, touch, contain, ... * Dimensionally Extended 9-Intersection Model (DE-9IM) * Decompose a MU into interior, boundary and exterior * Relationship base on the dimensionality of the intersection of boundary, interior, and exterior * 9-intersection model (9IM) is a special case (Egenhofer and Herring, 1991) ## DE-9IM * A feature decomposed into interior (I), boundary (B) and exterior (E) | Feature Type | Interior (I) | Boundary (B) | Exterior (E) | | :--- | :--- | :--- | :--- | | **Polygon** | points within the polygon | the rings | points not in the polygon | | **Line** | points of the line except endpoints | the endpoints | points not on the line | | **Point** | the point | empty set | points other than the point | * Dimensionality of intersection: {None (-1), Point (0), Line (1), Area (2)} ## DE-9IM Examples * A contains B: $I(B) \cap E(A) = \varnothing$ and $I(A) \cap I(B) \neq \varnothing$ * A disjoint B: $I(A) \cap I(B) = \varnothing$ and $I(A) \cap B(B) = \varnothing$ and $B(A) \cap I(B) = \varnothing$ and $B(A) \cap B(B) = \varnothing$ ## Quantitative Co-locational Relationship * Where is the shared/not shared space located? * Use mapping units (MUs) as the location * New MUs created as the result of overlapping existing MUs * Also called Map Overlay * Attributes of new MUs are associated with original attributes ## Map Overlay * Create new maps (MUs) based on existing maps (MUs) * Vertical integration * Two steps 1. Determine the geometry of new mapping units * Overlap existing mapping units 2. Assign attribute values to the new mapping units * Inherit from existing mapping units ## Geometry of New Mapping Units * Intersect boundaries of original MUs to create new MUs * New MUs are more "homogeneous" than original MUs ## Identify Relations between MUs * For two maps A and B, four kinds of new mapping units (at most) can be created: * MUs in A and also in B (A AND B) * MUs in A but not in B (A NOT B) * MUs in B but not in A (B NOT A) * MUs neither in A nor in B (NOT A AND NOT B) * Total space = (A AND B) + (A NOT B) + (B NOT A) + (NOT A AND NOT B) ## Map Overlay on Raster Maps * Determine the geometry of new MUs * Two raster maps must have the same "geometry" * Same resolution, same orientation, same extent * New MUs are just the original cells * Assign attribute values to the new MUs * Combine values of original cells at the same location * Map Algebra (Tomlin, 1990) * Relation between cells is 1-to-1 ## Map Overlay on Vector Maps * Determine the geometry of new MUs * Intersect boundaries of original MUs to create new MUs * Assign attribute values to the new MUs * Combine values of original MUs * Relation between MUs is typically M-to-N ## Attributes of the New Mapping Units * Typically a combination of the attributes of the original mapping units * Multi-valued map * $Value_{new} = \{Value_A, Value_B\}$ * A new attribute value can be calculated based on original values * $Value_{new} = f(Value_A, Value_B)$ * Boolean (0, 1), Arithmetic (+, -, *, /) ## Overlay Tools in ArcGIS * Different tools keep different kinds of new MUs * Intersect, Union, Identity, Erase, Symmetrical Difference, Update ## Map ArcGIS Overlay Tools to Kinds of MUs Kept * One kind of MUs * A AND B (intersect) * A NOT B (erase) * B NOT A (erase) * Two kinds of MUs * A NOT B, A AND B (identity) * B NOT A, A AND B (identity) * A NOT B and B NOT A (symmetrical difference) * All kinds of MUs * A AND B, A NOT B, B NOT A (union) ## Clip Tool in ArcGIS * What’s the difference between Intersect and Clip tool? * Attributes of the new MUs * Clip tool in the Extract toolset—Why? * How about Erase tool? ## Other Overlay Tools in ArcGIS * Two steps * Choose output MUs for the new map * Process/handle related MUs (including attributes) ## Overlay and Attribute Table * Number of new MUs (rows) * Vector: Usually increases * Raster: Stays the same * Number of attributes (columns) * Usually increases (attributes from both original maps) ## Dimensionally Extended Nine-Intersection Model (DE-9IM) **Author:** Christian Strobl, German Remote Sensing Data Center (DFD), German Aerospace Center (DLR) ## Synonyms Dimensionally Extended Nine-Intersection Model (DE-9IM), Nine-Intersection Model (9IM), Four-Intersection Model (4IM), Egenhofer Operators, Clementini Operators; Topological Operators. ## Definition The Dimensionally Extended Nine-Intersection Model (DE-9IM) or Clementini-Matrix is specified by the OGC "Simple Features for SQL" specification for computing the spatial relationships between geometries. It is based on the Nine-Intersection Model (9IM) or Egenhofer-Matrix, which is an extension of the Four-Intersection Model (4IM). The DE-9IM considers the two objects' interiors, boundaries, and exteriors and analyzes the intersections of these nine parts for their relationships. The model records the maximum dimension (-1, 0, 1, or 2) of the intersection geometries, where -1 corresponds to no intersection (empty set). The primary spatial relationships described by the DE-9IM are: * Equals * Disjoint * Intersects * Touches * Crosses * Within * Contains * Overlaps ## Main Text There are three common approaches for describing topological relationships between geodata, each based on an intersection matrix. ### Four-Intersection Model (4IM) The 4IM considers only the interior and the boundary of two objects. It creates a 2x2 matrix that describes whether the intersections between the interiors and boundaries of objects A and B are empty (0) or non-empty (1). ### Nine-Intersection Model (9IM) The 9IM extends the 4IM by including the exterior of the objects. This results in a 3x3 matrix. Like the 4IM, it traditionally uses boolean values (empty or non-empty) to describe the intersections of Interior (I), Boundary (B), and Exterior (E). ### Dimensionally Extended Nine-Intersection Model (DE-9IM) The DE-9IM further refines the 9IM by recording the **dimension** of the intersection rather than just a boolean state. The intersection of any two parts results in a set of points. The value in the DE-9IM matrix is the maximum dimension of this point set: * **-1**: The intersection is empty ($\emptyset$). * **0**: The intersection contains only points (0-dimensional). * **1**: The intersection contains lines (1-dimensional). * **2**: The intersection contains polygons/areas (2-dimensional). The matrix is structured as follows: | | Interior(B) | Boundary(B) | Exterior(B) | | :--- | :---: | :---: | :---: | | **Interior(A)** | dim(I(A)∩I(B)) | dim(I(A)∩B(B)) | dim(I(A)∩E(B)) | | **Boundary(A)** | dim(B(A)∩I(B)) | dim(B(A)∩B(B)) | dim(B(A)∩E(B)) | | **Exterior(A)** | dim(E(A)∩I(B)) | dim(E(A)∩B(B)) | dim(E(A)∩E(B)) | ## Geometric Operators and Predicates ### Disjoint $A \cap B = \emptyset$ The interiors and boundaries of the two geometries do not intersect at all. * Pattern: `FF*FF****` ### Intersects $A \cap B \neq \emptyset$ The complement of Disjoint. The geometries share at least one point. * Pattern: `T********` or `*T*******` or `***T*****` or `****T****` ### Touches $A \cap B \neq \emptyset$, but $Interior(A) \cap Interior(B) = \emptyset$. The geometries have at least one point in common, but their interiors do not overlap. This applies to Area/Area, Line/Line, Line/Area, and Point/Area, but not Point/Point. * Pattern: `FT*******` or `F**T*****` or `F***T****` ### Crosses The geometries share some but not all interior points, and the dimension of the intersection is less than the maximum dimension of the two geometries. * Pattern: `T*T******` (for Line/Area) or `0********` (for Line/Line) ### Within The geometry A lies entirely within the interior of B. * Pattern: `T*F**F***` ### Contains The inverse of Within. Geometry B is within Geometry A. * Pattern: `T*****FF*` ### Overlaps The geometries share some but not all points in common, and the intersection has the same dimension as the geometries themselves. * Pattern: `T*T***T**` (for Area/Area) or `1*T***T**` (for Line/Line) ### Equals The two geometries are topologically equal; they occupy the same space. * Pattern: `T*F**FFF*` ## Summary The DE-9IM is a powerful mathematical framework that allows GIS systems to query spatial relationships between complex objects rigorously. By using a 3x3 matrix of dimensions, it provides a unique "fingerprint" for every possible topological configuration between two geometric features.